ContinuousMonoidHom.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

Continuous Monoid Hom

Material destined for Mathlib.

Imports (7)

Imported by 2 files

def ContinuousAddEquiv.toIntContinuousLinearEquiv Reinterpret a continuous additive equivalence between additive groups as a continuous ℤ-linear equivalence line 27 source
Source documentation
Reinterpret a continuous additive equivalence between additive groups as a continuous ℤ-linear equivalence.

standard axioms: Quot.sound, propext what this means
Statement uses: AddCommGroup, TopologicalSpace, ContinuousAddEquiv, ContinuousLinearEquiv, Int, RingHom.id
Proof uses: LinearEquiv, AddEquiv.toIntLinearEquiv, ContinuousAddEquiv.toAddEquiv, ContinuousLinearEquiv.mk
18 instances, coercions and auxiliary constants left out.
```
def ContinuousAddEquiv.toIntContinuousLinearEquiv {M M₂ : Type*} [AddCommGroup M]
    [TopologicalSpace M] [AddCommGroup M₂] [TopologicalSpace M₂] (e : M ≃ₜ+ M₂) :
    M ≃L[ℤ] M₂ where
…
```
def ContinuousAddEquiv.quotientPi For a finite family of topological additive groups G i and subgroups p i ≤ G i, the canonical continuous additive isomorphism (∏ i, G i) ⧸ (∏ i, p i) ≃ₜ+ ∏ i, (G i ⧸ p i) line 37 source
Source documentation
For a finite family of topological additive groups G i and subgroups p i ≤ G i, the canonical continuous additive isomorphism (∏ i, G i) ⧸ (∏ i, p i) ≃ₜ+ ∏ i, (G i ⧸ p i).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, TopologicalSpace, IsTopologicalAddGroup, Fintype, AddSubgroup, DecidableEq, ContinuousAddEquiv, HasQuotient.Quotient, AddSubgroup.pi, Set.univ
Proof uses: ContinuousLinearEquiv.toContinuousAddEquiv, Int, Submodule, Submodule.pi, OrderIso, AddSubgroup.toIntSubmodule, Submodule.quotientPiContinuousLinearEquiv
34 instances, coercions and auxiliary constants left out.
```
def ContinuousAddEquiv.quotientPi {ι : Type*} {G : ι → Type*} [(i : ι) → AddCommGroup (G i)]
    [(i : ι) → TopologicalSpace (G i)]
    [(i : ι) → IsTopologicalAddGroup (G i)]
    [Fintype ι] (p : (i : ι) → AddSubgroup (G i)) [DecidableEq ι] :
    ((i : ι) → G i) ⧸ AddSubgroup.pi (_root_.Set.univ) p ≃ₜ+ ((i : ι) → G i ⧸ p i) :=
…
```
def ContinuousMulEquiv.piUnique line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Mul, TopologicalSpace, Unique, ContinuousMulEquiv, Inhabited.default
Proof uses: MulEquiv, MulEquiv.piUnique, ContinuousMulEquiv.mk
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def ContinuousMulEquiv.piEquivPiSubtypeProd line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Mul, DecidablePred, ContinuousMulEquiv, Prod, Subtype, Not
Proof uses: Homeomorph, Homeomorph.piEquivPiSubtypeProd, ContinuousMulEquiv.mk, MulEquiv.mk, Homeomorph.toEquiv
13 instances, coercions and auxiliary constants left out.
def ContinuousMulEquiv.mapUnits Any ContinuousMulEquiv induces a ContinuousMulEquiv on units line 69 source
Source documentation
Any ContinuousMulEquiv induces a ContinuousMulEquiv on units.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Monoid, ContinuousMulEquiv, Units
Proof uses: MulEquiv, Units.mapEquiv, ContinuousMulEquiv.mk
15 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def ContinuousMulEquiv.mapUnits {M N : Type*} [TopologicalSpace M] [TopologicalSpace N]
    [Monoid M] [Monoid N] (f : M ≃ₜ* N) : Mˣ ≃ₜ* Nˣ where
…
```
theorem ContinuousMulEquiv.coe_toHomeomorph line 76 source
standard axioms: Quot.sound what this means
Statement uses: TopologicalSpace, CommMonoid, ContinuousMulEquiv, Eq, Homeomorph, ContinuousMulEquiv.toHomeomorph
Proof uses: rfl
15 instances, coercions and auxiliary constants left out.